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Let $\mathcal{A}$ be an abelian category.

In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good candidate for "the" moduli stack of objects in $D^b(\mathcal{A})$.

You can also consider the category $D^b_2(\mathcal{A})$ of two periodic complexes in $\mathcal{A}$. In some senses it is meant to be "nicer" than $D^b(\mathcal{A})$, but I don't know enough about it to be more precise.

My question is whether there has been constructed a stack $X$ which is "the" moduli stack of objects in $D_2^b(\mathcal{A})$ - maybe perfect complexes on it recovers $D^b_2(\mathcal{A})$, or some other similar condition.

Then as a sanity check, when $\mathcal{A}=\text{Vect}^{f.d.}$ is finite dimensional vector spaces, what is this moduli stack? What is its cotangent complex, what is its cohomology?

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    $\begingroup$ Because the Ext groups are periodic, the homotopy groups of the stack won't be bounded, so it won't be locally truncated, hence not an Artin $\infty$-stack. (You can regard the problems as looking at $\mathcal{A}[y,y^{-1}]$-modules in complexes, for $y$ of degree $2$.) The derived deformation problem, along with other related ones, has been studied - see papers by Hua/Keller, Booth, ... $\endgroup$
    – Jon Pridham
    Commented Jan 26, 2023 at 15:48

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